# Darboux cyclides

## Overview

We show animations of Darboux cyclides together with some of their inversions.

A surface X is of **celestial type** (c,d,n) if

- the surface X contains c circles through almost each point,
- the degree of X is equal to d, and
- X is embedded into n-dimensional space and not contained in a hyperplane section.

The **singular type** (A,B) is a pair of
Dynkin types
where A denotes the real structure and B denotes the type of the isolated singularities.
The underlined Dynkin types correspond to real isolated singularities.
See here
for more information.
For the names of the cyclides we use the following abbreviations:

S = sphere, | N = non-rotational, |

E = elliptic/ellipsoid, | C = circular, |

H = hyperbolic/hyperboloid, | P = parabolic/paraboloid, |

1 = of one sheet, | 2 = of two sheets, |

O = cone, | Y = cylinder. |

A Darboux cyclide is either one of the following:

2S cyclide | HP cyclide |

1S cyclide | CH1 cyclide |

EH2 cyclide | spindle cyclide |

NE cyclide | ring cyclide |

EP cyclide | Perseus cyclide |

Blum cyclide | EY cyclide |

EH1 cyclide | horn cyclide |

EO cyclide |

The images were made using Surfex.

## 2S cyclide

Celestial type: (2,4,3)

Singular type: (D4,A0)

## 1S cyclide

Celestial type: (2,4,3)

Singular type: (3A1,A0)

## EH2 cyclide

Celestial type: (2,4,3)

Singular type: (3A1,__A1__)

## NE cyclide

Celestial type: (2,4,3)

Singular type: (3A1,__A1__)

## EP cyclide

Celestial type: (2,4,3)

Singular type: (3A1,__A2__)

## Blum cyclide

Celestial type: (6,4,3)

Singular type: (2A1,A0)

## EH1 cyclide

Celestial type: (4,4,3)

Singular type: (2A1,__A1__)

## EO cyclide

Celestial type: (3,4,3)

Singular type: (2A1,2__A1__)

## HP cyclide

Celestial type: (2,4,3)

Singular type: (2A1,__A2__)

## CH1 cyclide

Celestial type: (3,4,3)

Singular type: (2A1,__A1__+2A1)

## spindle cyclide

Celestial type: (3,4,3)

Singular type: (2A1,2__A1__+2A1)

a.k.a. CO cyclide

## ring cyclide

Celestial type: (4,4,3)

Singular type: (2A1,4A1)

## Perseus cyclide

Celestial type: (5,4,3)

Singular type: (2A1,2A1)

## EY cyclide

Celestial type: (3,4,3)

Singular type: (2A1,__A3__)

## horn cyclide

Celestial type: (3,4,3)

Singular type: (2A1,__A3__+2A1)

a.k.a. CY cyclide